For applications, it is crucial to have WZ-pairs at hand. In the previous work, WZ-pairs are obtained
either by guessing from the identities to be proved using Gosper's algorithm or by certain transformations from a given WZ-pair [1]}.
Riordan in the preface of his book [2]} commented that “the central fact developed is that identities are both
inexhaustible and unpredictable; the age-old dream of putting order in this chaos is doomed to failure ”. As an optimistic respond to Riordan's comment,
Gessel in his talkThe talk was given at the Waterloo Workshop in Computer Algebra (in honor of Herbert Wilf's 80th birthday), Wilfrid Laurier University, May 28, 2011.
For the talk slides, see the link: http://people.brandeis.edu/~gessel/homepage/slides/wilf80-slides.pdf on the WZ method motivated with some examples that “WZ forms bring order to this chaos ”, where WZ-forms are a multivariate generalization of WZ-pairs [3]}. With the hope of discovering more combinatorial identities in an intrinsic and algorithmic way, it is natural and challenging to ask the following question.
